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Topic: Algorithm for effective decidability
Replies: 4   Last Post: Oct 1, 2017 4:24 PM

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Mike Terry

Posts: 767
Registered: 12/6/04
Re: Algorithm for effective decidability
Posted: Sep 29, 2017 6:09 PM
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On 29/09/2017 21:47, agapito wrote:
> Assume 's' is in set of numbers S, and consider the algorithm which, for input 'n', effectively tests whether 'n' is in S. If it is, algorithm gets a 'yes' and ouputs 'n', otherwise outputs 's'. My problem understanding this is that obviously if 'n' is not in S, then the algorithm will churn forever, so it's not clear under what circumstance it would ouput s. Does every such algorithm contain an internal limit on how long it can continue, as it would require under actual computation? Or in theoretical treatment it can, indeed run forever?
> All clarifications appreciated, am

An effective test of whether n is in S must terminate and indicate
whether OR NOT n is in S. In your example, if n is in S we "get a yes",
so I guess if n is not in S we "get a no". (I.e. there must be two
halting states, one for 'yes' and another for 'no'.)

You are right concerning the output from the algorithm - for the output
to be defined, the algorithm must halt.

If a set S has the weaker property that the algorithm will "get a yes"
for input n if and only if n is in S, (and otherwise may "get something
else" or never terminate) then the set S is generally called
(effectively) semi-decidable, rather than decidable.


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